Series expansions for the electric and magnetic fields produced by a line or sheet current source above a layered Earth

Pirjola, Risto; Viljanen, A.; Boteler, D. H.

doi:10.1029/1998rs900034

Cited by 12 publications

(8 citation statements)

References 10 publications

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“…By looking at –, Pirjola et al [1999] concluded that similar series expansions are also obtained for a layered Earth if p can be expressed as

where the “equivalent propagation constant” k u is independent of b . A comparison of and defines the “equivalent surface impedance” Z u :

Demanding that Z u and Z are equal when b = 0,

where Z p = Z ( b = 0) is the "plane wave surface impedance."…”

Section: Series Expansionsmentioning

confidence: 78%

“…It should be noted that p equals the “inductive response function” used in geoelectromagnetic induction studies of the Earth [ Schmucker , 1970]. Substituting c into formulas –, the electric and magnetic fields also have the following EXA expressions [ Pirjola et al , 1999]:

…”

Section: Modelmentioning

confidence: 99%

“…When b goes to infinity, the correct layered‐Earth surface impedance approaches i ωμ 0 / b , and this is also the limit of Z u . Consequently, Z and Z u only differ for a range of small (nonzero) b values [ Pirjola et al , 1999]. For a uniform Earth, Z and Z u , as well as k and k u , are equal (, , , and ).…”

Section: Series Expansionsmentioning

confidence: 99%

“…Recently, exact series expansions have been presented by Pirjola [1998] for both the electric and magnetic fields on a uniform Earth. Approximate series expansions may also be derived in the case of a layered Earth [ Pirjola et al , 1999].…”

Section: Introductionmentioning

confidence: 99%

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Calculation methods of the electric and magnetic fields at the Earth's surface produced by a line current

Pirjola

Boteler

2002

Radio Science

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[1] Space weather storms involve intense and rapidly varying electric currents in the ionosphere, which create electric and magnetic fields at the Earth's surface. The electric fields drive geomagnetically induced currents (GIC) in technological networks and may have serious impacts. For assessing the hazards it is necessary to estimate GIC magnitudes, and this requires calculations of the electric and magnetic fields produced at the Earth's surface by the ionospheric currents. The surface fields are also affected by currents induced within the ground and influenced by the conductivity of the Earth. This also has to be taken into account. The calculation methods should be fast enough that they can be applied to forecasting the fields and GIC, for example, by using satellite observations of the solar wind. In this paper, we consider an infinitely long horizontal line current, which is the basic model of an auroral electrojet and simple enough to give insight into the physics and calculation techniques. The Earth is assumed to be composed of horizontal layers. We consider the exact integral expressions of the fields at the Earth's surface. The applicability of a series expansion technique (SER) and the complex image method (CIM), both of which were originally developed for other disciplines, are reviewed and summarized by giving the expressions of the electric and magnetic fields at the Earth's surface and by considering the mathematical assumptions required. Numerical calculations and comparisons with exact solutions show that SER and CIM are very accurate.

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“…By looking at –, Pirjola et al [1999] concluded that similar series expansions are also obtained for a layered Earth if p can be expressed as

where the “equivalent propagation constant” k u is independent of b . A comparison of and defines the “equivalent surface impedance” Z u :

Demanding that Z u and Z are equal when b = 0,

where Z p = Z ( b = 0) is the "plane wave surface impedance."…”

Section: Series Expansionsmentioning

confidence: 78%

…”

Section: Modelmentioning

confidence: 99%

Section: Series Expansionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Calculation methods of the electric and magnetic fields at the Earth's surface produced by a line current

Pirjola

Boteler

2002

Radio Science

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“…Nabulsi and Wait [7] obtained the transient response of a two-layer earth model. Pirjola [8] and Pirjola et al [9] demonstrated that the fields at the surface of a layered earth can be expressed as approximate series expansions analogous to the exact series valid for a uniform earth due to a line current.…”

Section: Introductionmentioning

confidence: 98%

Transient behaviour of magnetic and electric fields at any height above an atmospheric duct

Abo-Seida¹,

Pirjola²

2008

Can. J. Phys.

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We model the atmosphere by a surface duct underlying a half-space. A derivation is given for the electromagnetic field created by a vertical magnetic dipole, in the half-space above the duct, with an arbitrary time-varying moment. The method used for the solution is essentially based on the application of two functional transforms. Starting from the wave equation for the magnetic field and applying a Laplace transform in time, we obtain a two-dimensional Fourier transform in the horizontal spatial coordinates leading to an integral representation of the solution of the wave equation in the transform space. The transient behavior of the magnetic-field strength at any distance above the duct is determined. PACS Nos.: 41.20Jb, 42.25Bs, 42.25Gy Résumé : Nous avons modélisé l'atmosphère par un conduit de surface sous-jacent à un demi-espace. Nous obtenons une expression pour le champ électromagnétique dans le demi-espace supérieur, créé par un dipôle magnétique vertical dont le moment a une variation temporelle arbitraire. La méthode de solution utilisée est basée essentiellement sur l'utilisation de deux transformations fonctionnelles. Débutant avec l'équation d'onde pour le champ magnétique, nous utilisons une transformation de Laplace sur le temps et appliquons ensuite une transformation de Fourier sur les coordonnées spatiales horizontales, ce qui mène à une représentation intégrale de la solution de l'équation d'onde dans l'espace transformé. Nous avons déterminé le comportement transitoire du champ magnétique pour toute distance au dessus du conduit.[Traduit par la Rédaction]

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Proximity Effect on the Induced Geoelectric Field at the Lateral Interface of Different Conductivity Structures During Geomagnetic Storms

Dong

Wang

Liu

et al. 2015

Chinese J of Geophysics

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The induced geoelectric fields and telluric currents generated during geomagnetic storms will be distorted by lateral variations of the Earth conductivity. The Galerkin finite element method (FEM) is a useful tool to analyze complicated electromagnetic field problems. In this paper, a uniform thin sheet current with infinite width located at 100 km above the Earth's surface is assumed to represent the source of geomagnetic variations. It is harmonic with the amplitude of 1 A/m at different frequencies. Three Earth conductivity structures are modelled where the basic model is horizontally uniform for comparisons. The second and third structures have the same exact thicknesses and conductivity values with the basic one on one side and five times greater or lower values on the other side to represent conductivity variation from low to high and from high to low. The computational results show that the abrupt changes of conductivities have influences on the induced telluric currents, which can be classified as the proximity effect and the skin effect. For GIC research in power systems, especially in systems located at coastal areas, the geoelectric field and GIC should be determined taking the directions of the transmission lines and of the coast into consideration. Both the proximity effect and the skin effect should be considered along with the coast effect in research of GIC impacts during geomagnetic disturbances. Boundary conditions discussed in this paper can be varied depending on different modelling techniques and on the scales of conductivity structure models.

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Series expansions for the electric and magnetic fields produced by a line or sheet current source above a layered Earth

Cited by 12 publications

References 10 publications

Calculation methods of the electric and magnetic fields at the Earth's surface produced by a line current

Calculation methods of the electric and magnetic fields at the Earth's surface produced by a line current

Transient behaviour of magnetic and electric fields at any height above an atmospheric duct

Proximity Effect on the Induced Geoelectric Field at the Lateral Interface of Different Conductivity Structures During Geomagnetic Storms

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